Differential equations: a dynamical approach for the Poincaré-Hopf Theorem

• Grant number: 21/04961-1
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): June 01, 2021
• Effective date (End): May 31, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Nivaldo de Góes Grulha Júnior
• Grantee: Luís André Mucheroni Covolan
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/21181-0 - New frontiers in Singularity Theory, AP.TEM

Abstract

The study of vector fields in the vicinity of asingularity, or stationary point, is a widely studied topic due to its importance in various branches of mathematics and its applications in other areas of science, such as biology, meteorology and etc. Often, when we want to equate certainphenomena, we are faced with equations that involve variations in certain quantities deemed essential. The use of equations differentials to represent these variations is essentially due to the fact that the studied phenomenon is seen in discrete time or continuous.Because many differential equations are notconveniently soluble by analytical methods, it is important to consider qualitative information obtained from their solutions, without indeed solve them. We will see how this can be done. Thus, some methods of solving differential equations are presented most important or best known. We will also focus onlinear systems, as this theory will be used in the qualitative study of non-linear systems.Information about the behavior of a nearby vector field of a singular point can be obtained through some invariants. O most basic invariant of a vector field at a singular point is called the Poincaré-Hopf Index. Very important concept, and considered a true link between mathematical branchessuch as Algebraic Topology and Differential Topology. Such a connection is possible through the important result known as Poincaré-Hopf Theorem, the final objective of this project.